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G = C11×D16order 352 = 25·11

Direct product of C11 and D16

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Aliases: C11×D16, C1765C2, C161C22, D81C22, C22.15D8, C44.36D4, C88.24C22, (C11×D8)⋊5C2, C8.2(C2×C22), C4.1(D4×C11), C2.3(C11×D8), SmallGroup(352,60)

Series: Derived Chief Lower central Upper central

C1C8 — C11×D16
C1C2C4C8C88C11×D8 — C11×D16
C1C2C4C8 — C11×D16
C1C22C44C88 — C11×D16

Generators and relations for C11×D16
 G = < a,b,c | a11=b16=c2=1, ab=ba, ac=ca, cbc=b-1 >

8C2
8C2
4C22
4C22
8C22
8C22
2D4
2D4
4C2×C22
4C2×C22
2D4×C11
2D4×C11

Smallest permutation representation of C11×D16
On 176 points
Generators in S176
(1 164 31 111 67 39 123 129 153 58 96)(2 165 32 112 68 40 124 130 154 59 81)(3 166 17 97 69 41 125 131 155 60 82)(4 167 18 98 70 42 126 132 156 61 83)(5 168 19 99 71 43 127 133 157 62 84)(6 169 20 100 72 44 128 134 158 63 85)(7 170 21 101 73 45 113 135 159 64 86)(8 171 22 102 74 46 114 136 160 49 87)(9 172 23 103 75 47 115 137 145 50 88)(10 173 24 104 76 48 116 138 146 51 89)(11 174 25 105 77 33 117 139 147 52 90)(12 175 26 106 78 34 118 140 148 53 91)(13 176 27 107 79 35 119 141 149 54 92)(14 161 28 108 80 36 120 142 150 55 93)(15 162 29 109 65 37 121 143 151 56 94)(16 163 30 110 66 38 122 144 152 57 95)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 28)(18 27)(19 26)(20 25)(21 24)(22 23)(29 32)(30 31)(33 44)(34 43)(35 42)(36 41)(37 40)(38 39)(45 48)(46 47)(49 50)(51 64)(52 63)(53 62)(54 61)(55 60)(56 59)(57 58)(65 68)(66 67)(69 80)(70 79)(71 78)(72 77)(73 76)(74 75)(81 94)(82 93)(83 92)(84 91)(85 90)(86 89)(87 88)(95 96)(97 108)(98 107)(99 106)(100 105)(101 104)(102 103)(109 112)(110 111)(113 116)(114 115)(117 128)(118 127)(119 126)(120 125)(121 124)(122 123)(129 144)(130 143)(131 142)(132 141)(133 140)(134 139)(135 138)(136 137)(145 160)(146 159)(147 158)(148 157)(149 156)(150 155)(151 154)(152 153)(161 166)(162 165)(163 164)(167 176)(168 175)(169 174)(170 173)(171 172)

G:=sub<Sym(176)| (1,164,31,111,67,39,123,129,153,58,96)(2,165,32,112,68,40,124,130,154,59,81)(3,166,17,97,69,41,125,131,155,60,82)(4,167,18,98,70,42,126,132,156,61,83)(5,168,19,99,71,43,127,133,157,62,84)(6,169,20,100,72,44,128,134,158,63,85)(7,170,21,101,73,45,113,135,159,64,86)(8,171,22,102,74,46,114,136,160,49,87)(9,172,23,103,75,47,115,137,145,50,88)(10,173,24,104,76,48,116,138,146,51,89)(11,174,25,105,77,33,117,139,147,52,90)(12,175,26,106,78,34,118,140,148,53,91)(13,176,27,107,79,35,119,141,149,54,92)(14,161,28,108,80,36,120,142,150,55,93)(15,162,29,109,65,37,121,143,151,56,94)(16,163,30,110,66,38,122,144,152,57,95), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(29,32)(30,31)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(45,48)(46,47)(49,50)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(65,68)(66,67)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88)(95,96)(97,108)(98,107)(99,106)(100,105)(101,104)(102,103)(109,112)(110,111)(113,116)(114,115)(117,128)(118,127)(119,126)(120,125)(121,124)(122,123)(129,144)(130,143)(131,142)(132,141)(133,140)(134,139)(135,138)(136,137)(145,160)(146,159)(147,158)(148,157)(149,156)(150,155)(151,154)(152,153)(161,166)(162,165)(163,164)(167,176)(168,175)(169,174)(170,173)(171,172)>;

G:=Group( (1,164,31,111,67,39,123,129,153,58,96)(2,165,32,112,68,40,124,130,154,59,81)(3,166,17,97,69,41,125,131,155,60,82)(4,167,18,98,70,42,126,132,156,61,83)(5,168,19,99,71,43,127,133,157,62,84)(6,169,20,100,72,44,128,134,158,63,85)(7,170,21,101,73,45,113,135,159,64,86)(8,171,22,102,74,46,114,136,160,49,87)(9,172,23,103,75,47,115,137,145,50,88)(10,173,24,104,76,48,116,138,146,51,89)(11,174,25,105,77,33,117,139,147,52,90)(12,175,26,106,78,34,118,140,148,53,91)(13,176,27,107,79,35,119,141,149,54,92)(14,161,28,108,80,36,120,142,150,55,93)(15,162,29,109,65,37,121,143,151,56,94)(16,163,30,110,66,38,122,144,152,57,95), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(29,32)(30,31)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(45,48)(46,47)(49,50)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(65,68)(66,67)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88)(95,96)(97,108)(98,107)(99,106)(100,105)(101,104)(102,103)(109,112)(110,111)(113,116)(114,115)(117,128)(118,127)(119,126)(120,125)(121,124)(122,123)(129,144)(130,143)(131,142)(132,141)(133,140)(134,139)(135,138)(136,137)(145,160)(146,159)(147,158)(148,157)(149,156)(150,155)(151,154)(152,153)(161,166)(162,165)(163,164)(167,176)(168,175)(169,174)(170,173)(171,172) );

G=PermutationGroup([(1,164,31,111,67,39,123,129,153,58,96),(2,165,32,112,68,40,124,130,154,59,81),(3,166,17,97,69,41,125,131,155,60,82),(4,167,18,98,70,42,126,132,156,61,83),(5,168,19,99,71,43,127,133,157,62,84),(6,169,20,100,72,44,128,134,158,63,85),(7,170,21,101,73,45,113,135,159,64,86),(8,171,22,102,74,46,114,136,160,49,87),(9,172,23,103,75,47,115,137,145,50,88),(10,173,24,104,76,48,116,138,146,51,89),(11,174,25,105,77,33,117,139,147,52,90),(12,175,26,106,78,34,118,140,148,53,91),(13,176,27,107,79,35,119,141,149,54,92),(14,161,28,108,80,36,120,142,150,55,93),(15,162,29,109,65,37,121,143,151,56,94),(16,163,30,110,66,38,122,144,152,57,95)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,28),(18,27),(19,26),(20,25),(21,24),(22,23),(29,32),(30,31),(33,44),(34,43),(35,42),(36,41),(37,40),(38,39),(45,48),(46,47),(49,50),(51,64),(52,63),(53,62),(54,61),(55,60),(56,59),(57,58),(65,68),(66,67),(69,80),(70,79),(71,78),(72,77),(73,76),(74,75),(81,94),(82,93),(83,92),(84,91),(85,90),(86,89),(87,88),(95,96),(97,108),(98,107),(99,106),(100,105),(101,104),(102,103),(109,112),(110,111),(113,116),(114,115),(117,128),(118,127),(119,126),(120,125),(121,124),(122,123),(129,144),(130,143),(131,142),(132,141),(133,140),(134,139),(135,138),(136,137),(145,160),(146,159),(147,158),(148,157),(149,156),(150,155),(151,154),(152,153),(161,166),(162,165),(163,164),(167,176),(168,175),(169,174),(170,173),(171,172)])

121 conjugacy classes

class 1 2A2B2C 4 8A8B11A···11J16A16B16C16D22A···22J22K···22AD44A···44J88A···88T176A···176AN
order122248811···111616161622···2222···2244···4488···88176···176
size11882221···122221···18···82···22···22···2

121 irreducible representations

dim111111222222
type++++++
imageC1C2C2C11C22C22D4D8D16D4×C11C11×D8C11×D16
kernelC11×D16C176C11×D8D16C16D8C44C22C11C4C2C1
# reps112101020124102040

Matrix representation of C11×D16 in GL2(𝔽353) generated by

1310
0131
,
150340
183163
,
150340
20203
G:=sub<GL(2,GF(353))| [131,0,0,131],[150,183,340,163],[150,20,340,203] >;

C11×D16 in GAP, Magma, Sage, TeX

C_{11}\times D_{16}
% in TeX

G:=Group("C11xD16");
// GroupNames label

G:=SmallGroup(352,60);
// by ID

G=gap.SmallGroup(352,60);
# by ID

G:=PCGroup([6,-2,-2,-11,-2,-2,-2,553,3171,1593,165,7924,3970,88]);
// Polycyclic

G:=Group<a,b,c|a^11=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C11×D16 in TeX

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